Modelling with Differential and Difference Equations

418 Pages·1997·4.585 MB·English

by  Glenn Fulford, Peter Forrester, Arthur Jones

Australian Mathematical Society Lectures Series L(
Modelling with
Differential and
Difference Equations
GLENN FULFORD;
TETER FORRESTER
ARTHUR, JONESP
Modelling with Differential
and Difference Equations
AUSTRALIAN MATHEMATICAL SOCIETY LECTURE SERIES
Editor-in-chief: Professor J.H. Loxton, School of Mathematics, Physics,
Computing and Electronics, Macquarie University, NSW 2109,
Australia
1 Introduction to Linear and Convex Programming, N. CAMERON
2 Manifolds and Mechanics, A. JONES, A. GRAY & R. HUTTON
3 Introduction to the Analysis of Metric Spaces, J. R. GILES
4 An Introduction to Mathematical Physiology and Biology, J. MAZUMDAR
5 2-Knots and their Groups, J. HILLMAN
6 The Mathematics of Projectiles in Sport, N. DE MESTRE
7 The Peterson Graph, D. A. HOLTON & J. SHEEHAN
8 Low Rank Representations and Graphs for Sporadic Groups,
C. PRAEGAR & L. SOICHER
9 Algebraic Groups and Lie Groups, G. LEHRER (ed)
Modelling with Differential
and Difference Equations
GLENN FULFORD
Department of Mathematics,
University College ADFA, Canberra
PETER FORRESTER
Department of Mathematics,
Melbourne University
ARTHUR JONES
Department of Mathematics,
Latrobe University
CAMBRIDGE
UNIVERSITY PRESS
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, SAo Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 2RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521440691
© Cambridge University Press 1997
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 1997
Reprinted 2001
A catalogue record for this publication is available from the British Library
ISBN-13 978-0-521-44069-1 hardback
ISBN-10 0-521-44069-6 hardback
ISBN-13 978-0-521-44618-1 paperback
ISBN- 10 0-521-44618-X paperback
Transferred to digital printing 2006
Contents
Preface
Introduction to the student
Part one: Simple Models in Mechanics
Newtonian mechanics
1
1.1 Mechanics before Newton
1.2 Kinematics and dynamics
1.3 Newton's laws
1.4 Gravity near the Earth
1.5 Units and dimensions
2 Kinematics on a line
2.1 Displacement and velocity
2.2 Acceleration
2.3 Derivatives as slopes
2.4 Differential equations and antiderivatives
3 Ropes and pulleys
3.1 Tension in the rope
3.2 Solving pulley problems
3.3 Further pulley systems
3.4 Symmetry
4 Friction
4.1 Coefficients of friction
4.2 Further applications
4.3 Why does the wheel work?
5 Differential equations: linearity and SHM
5.1 Guessing solutions
5.2 How many solutions?
5.3 Linearity
5.4 The SHM equation
v
vi Contents
6 Springs and oscillations 85
6.1 Force in a spring 85
6.2 A basic example 88
6.3 Further spring problems 94
Part two: Models with Difference Equations 103
7 Difference equations 105
7.1 Introductory example 105
7.2 Difference equations basic ideas 109
7.3 Constant solutions and fixed points 114
7.4 Iteration and cobweb diagrams 118
8 Linear difference equations in finance and economics 126
8.1 Linearity 127
8.2 Interest and loan repayment 133
8.3 The cobweb model of supply and demand 138
8.4 National income : `acceleration models' 142
9 Non-linear difference equations and population growth 146
9.1 Linear models for population growth 146
9.2 Restricted growth non-linear models 152
9.3 A computer experiment 157
9.4 A coupled model of a measles epidemic 164
9.5 Linearizing non-linear equations 170
10 Models for population genetics 177
10.1 Some background genetics 177
10.2 Random mating with equal survival 185
10.3 Lethal recessives, selection and mutation 193
Part three: Models with Differential Equations 201
11 Continuous growth and decay models 203
11.1 First-order differential equations 203
11.2 Exponential growth 212
11.3 Restricted growth 218
11.4 Exponential decay 227
12 Modelling heat flow 232
12.1 Newton's model of heating and cooling 232
12.2 More physics in the model 237
12.3 Conduction and insulation 241
12.4 Insulating a pipe 249
13 Compartment models of mixing 257
13.1 A mixing problem 257
13.2 Modelling pollution in a lake 265
13.3 Modelling heat loss from a hot water tank 270
Contents vii
Part four: Further Mechanics 275
14 Motion in a fluid medium 277
14.1 Some basic fluid mechanics 277
14.2 Archimedes' Principle 282
14.3 Falling sphere with Stokes' resistance 286
14.4 Falling sphere with velocity-squared drag 290
15 Damped and forced oscillations 295
15.1 Constant-coefficient differential equations 295
15.2 Damped oscillations 302
15.3 Forced harmonic motion 311
16 Motion in a plane 318
16.1 Kinematics in a plane 318
16.2 Motion down an inclined plane 326
16.3 Projectiles 331
17 Motion on a circle 336
17.1 Kinematics on a circle 336
17.2 Uniform circular motion 343
17.3 The pendulum and linearization 348
Part five: Coupled Models 353
18 Models with linear interactions 355
18.1 Two-compartment mixing 355
18.2 Solving constant-coefficient equations 360
18.3 A model for detecting diabetes 366
18.4 Nutrient exchange in the placenta 373
19 Non-linear coupled models 379
19.1 Predator-prey interactions 379
19.2 Phase-plane analysis 384
19.3 Models of combat 389
19.4 Epidemics 394
References 399
Index 403

Preface
This book provides an introduction to modelling with both differential
and difference equations. Our approach to mathematical modelling is
to emphasize what is involved by looking at specific examples from
a variety of disciplines. From each discipline enough background is
provided to enable students to understand both the assumptions and the
predictions of the models. Exercises have been included at the end of
each section. They are intended to provide a balanced development of
some of the main skills used in mathematical modelling, and hence they
are an essential part of the book.
The main mathematical tools used in the book are differential and dif-
ference equations. Differential equations have their origins in mechanics :
Newton's laws of motion lead to differential equations whose solutions
can be used to predict the position of a body at some later time. Dif-
ferential equations have been closely associated with the rise of physical
science in previous centuries and they are now being used as models for
real world problems in a variety of other disciplines. Difference equations
are the discrete analogues of differential equations. They have risen
to prominence in the last decade, during which it has become generally
known that solutions of even very simple difference equations can exhibit
complex chaotic behaviour.
To allow time for the development of other modelling skills besides
solving the equations arising from the models, we have selected only
models involving differential equations which are relatively easy to solve.
Although our treatment of differential equations is intended to be self-
contained, it is only fair to point out that our students were taking
concurrently a first course in mathematical methods (beginning with
the elements of differentiation and ending with some practice at solving
separable and linear constant-coefficient differential equations, towards
ix